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2012

Volume 50, Issue 1, pp. 1-372


Convergence of a Particle Method and Global Weak Solutions of a Family of Evolutionary PDEs

Alina Chertock, Jian-Guo Liu, and Terrance Pendleton

SIAM J. Numer. Anal. 50, pp. 1-21 (21 pages)

Online Publication Date: January 19, 2012

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The purpose of this paper is to provide global existence and uniqueness results for a family of fluid transport equations by establishing convergence results for the particle method applied to these equations. The considered family of PDEs is a collection of strongly nonlinear equations which yield traveling wave solutions and can be used to model a variety of flows in fluid dynamics. We apply a particle method to the studied evolutionary equations and provide a new self-contained method for proving its convergence. The latter is accomplished by using the concept of space-time bounded variation and the associated compactness properties. From this result, we prove the existence of a unique global weak solution in some special cases and obtain stronger regularity properties of the solution than previously established.

A Computational Measure Theoretic Approach to Inverse Sensitivity Problems II: A Posteriori Error Analysis

T. Butler, D. Estep, and J. Sandelin

SIAM J. Numer. Anal. 50, pp. 22-45 (24 pages) | Cited 1 time

Online Publication Date: January 19, 2012

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In part one of this paper [T. Butler and D. Estep, SIAM J. Numer. Anal., to appear], we develop and analyze a numerical method to solve a probabilistic inverse sensitivity analysis problem for a smooth deterministic map assuming that the map can be evaluated exactly. In this paper, we treat the situation in which the output of the map is determined implicitly and is difficult and/or expensive to evaluate, e.g., requiring the solution of a differential equation, and hence the output of the map is approximated numerically. The main goal is an a posteriori error estimate that can be used to evaluate the accuracy of the computed distribution solving the inverse problem, taking into account all sources of statistical and numerical deterministic errors. We present a general analysis for the method and then apply the analysis to the case of a map determined by the solution of an initial value problem.

A State Space Error Estimate for POD-DEIM Nonlinear Model Reduction

Saifon Chaturantabut and Danny C. Sorensen

SIAM J. Numer. Anal. 50, pp. 46-63 (18 pages)

Online Publication Date: January 19, 2012

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This paper derives state space error bounds for the solutions of reduced systems constructed using proper orthogonal decomposition (POD) together with the discrete empirical interpolation method (DEIM) recently developed for nonlinear dynamical systems [SIAM J. Sci. Comput., 32 (2010), pp. 2737–2764]. The resulting error estimates are shown to be proportional to the sums of the singular values corresponding to neglected POD basis vectors both in Galerkin projection of the reduced system and in the DEIM approximation of the nonlinear term. The analysis is particularly relevant to ODE systems arising from spatial discretizations of parabolic PDEs. The derivation clearly identifies where the parabolicity is crucial. It also explains how the DEIM approximation error involving the nonlinear term comes into play.

Convergence of Hill's Method for Nonselfadjoint Operators

Mathew A. Johnson and Kevin Zumbrun

SIAM J. Numer. Anal. 50, pp. 64-78 (15 pages)

Online Publication Date: January 19, 2012

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By the introduction of a generalized Evans function defined by an appropriate $2$-modified Fredholm determinant, we give a simple proof of convergence in location and multiplicity of Hill's method for numerical approximation of spectra of periodic-coefficient ordinary differential operators. Our results apply to operators of nondegenerate type under the condition that the principal coefficient matrix be symmetric positive definite (automatically satisfied in the scalar case). Notably, this includes a large class of non-self-adjoint operators which previously had not been treated in a simple way. The case of general coefficients depends on an interesting operator-theoretic question regarding properties of Toeplitz matrices.

Optimal Error Estimates of the Semidiscrete Local Discontinuous Galerkin Methods for High Order Wave Equations

Yan Xu and Chi-Wang Shu

SIAM J. Numer. Anal. 50, pp. 79-104 (26 pages)

Online Publication Date: January 19, 2012

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In this paper, we introduce a general approach for proving optimal $L^2$ error estimates for the semidiscrete local discontinuous Galerkin (LDG) methods solving linear high order wave equations. The optimal order of error estimates holds not only for the solution itself but also for the auxiliary variables in the LDG method approximating the various order derivatives of the solution. Examples including the one-dimensional third order wave equation, one-dimensional fifth order wave equation, and multidimensional Schrödinger equation are explored to demonstrate this approach. The main idea is to derive energy stability for the various auxiliary variables in the LDG discretization by using the scheme and its time derivatives with different test functions. Special projections are utilized to eliminate the jump terms at the cell boundaries in the error estimate in order to achieve the optimal order of accuracy.

Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich–Schwoebel Type Energy: Application to Thin Film Epitaxy

Jie Shen, Cheng Wang, Xiaoming Wang, and Steven M. Wise

SIAM J. Numer. Anal. 50, pp. 105-125 (21 pages)

Online Publication Date: January 26, 2012

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We construct unconditionally stable, unconditionally uniquely solvable, and second-order accurate (in time) schemes for gradient flows with energy of the form $\int_\Omega ( F(\nabla\phi({\bf x})) + \frac{\epsilon^2}{2}|\Delta\phi({\bf x})|^2 ) d{\bf x}$. The construction of the schemes involves the appropriate combination and extension of two classical ideas: (i) appropriate convex-concave decomposition of the energy functional and (ii) the secant method. As an application, we derive schemes for epitaxial growth models with slope selection ($F({\bf y})= \frac14(|{\bf y}|^2-1)^2$) or without slope selection ($F({\bf y})=-\frac12\ln(1+|{\bf y}|^2)$). Two types of unconditionally stable uniquely solvable second-order schemes are presented. The first type inherits the variational structure of the original continuous-in-time gradient flow, while the second type does not preserve the variational structure. We present numerical simulations for the case with slope selection which verify well-known physical scaling laws for the long time coarsening process.

Long Time Stability of a Classical Efficient Scheme for Two-dimensional Navier–Stokes Equations

S. Gottlieb, F. Tone, C. Wang, X. Wang, and D. Wirosoetisno

SIAM J. Numer. Anal. 50, pp. 126-150 (25 pages)

Online Publication Date: January 26, 2012

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This paper considers the long-time stability property of a popular semi-implicit scheme for the two-dimensional incompressible Navier–Stokes equations in a periodic box that treats the viscous term implicitly and the nonlinear advection term explicitly. We consider both the semidiscrete (discrete in time but continuous in space) and fully discrete schemes with either Fourier Galerkin spectral or Fourier pseudospectral (collocation) methods. We prove that in all cases, the scheme is long time stable provided that the timestep is sufficiently small. The long time stability in the $L^2$ and $H^1$ norms further leads to the convergence of the global attractors and invariant measures of the scheme to those of the Navier–Stokes equations at vanishing timestep.

Robust Equilibrated Residual Error Estimator for Diffusion Problems: Conforming Elements

Zhiqiang Cai and Shun Zhang

SIAM J. Numer. Anal. 50, pp. 151-170 (20 pages)

Online Publication Date: February 02, 2012

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This paper analyzes an equilibrated residual a posteriori error estimator for the diffusion problem. The estimator, which is a modification of those in [D. Braess and J. Schöberl, Math. Comput., 77 (2008), pp. 651–672; R. Verfürth, SIAM J. Numer. Anal., 47 (2009), pp. 3180–3194], is based on the Prager–Synge identity and on a local recovery of an equilibrated flux. Numerical results for an interface test problem show that the modification is necessary for the robustness of the estimator. When the distribution of diffusion coefficients is local quasi-monotone, it is shown theoretically that the estimator is robust with respect to the size of jumps.

Matrix Probing and its Conditioning

Jiawei Chiu and Laurent Demanet

SIAM J. Numer. Anal. 50, pp. 171-193 (23 pages)

Online Publication Date: February 09, 2012

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When a matrix $A$ with $n$ columns is known to be well-approximated by a linear combination of basis matrices $B_1,\ldots,B_p$, we can apply $A$ to a random vector and solve a linear system to recover this linear combination. The same technique can be used to obtain an approximation to $A^{-1}$. A basic question is whether this linear system is well-conditioned. This is important for two reasons: a well-conditioned system means (1) we can invert it and (2) the error in the reconstruction can be controlled. In this paper, we show that if the Gram matrix of the $B_j$'s is sufficiently well-conditioned and each $B_j$ has a high numerical rank, then $n\propto p \log^2 n$ will ensure that the linear system is well-conditioned with high probability. Our main application is probing linear operators with smooth pseudodifferential symbols such as the wave equation Hessian in seismic imaging [L. Demanet et al., Appl. Comput. Harmonic Anal., 32 (2012), pp. 155–168]. We also demonstrate numerically that matrix probing can produce good preconditioners for inverting elliptic operators in variable media.

Finite Element Approximation of Elliptic Problems with Dirac Measure Terms in Weighted Spaces: Applications to One- and Three-dimensional Coupled Problems

Carlo D'Angelo

SIAM J. Numer. Anal. 50, pp. 194-215 (22 pages)

Online Publication Date: February 09, 2012

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In this work we study the stability and the convergence rates of the finite element approximation of elliptic problems involving Dirac measures, using weighted Sobolev spaces and weighted discrete norms. Our approach handles both the cases where the measure is simply a right-hand side and where it represents an additional term, i.e., solution-dependent, in the formulation of the problem. The main motivation of this study is to provide a methodological tool to treat elliptic problems in fractured domains, where the coupling terms are seen as Dirac measures concentrated on the fractures. We first establish a decomposition lemma, which is our fundamental tool for the analysis of the considered problems in the nonstandard setting of weighted spaces. Then, we consider the stability of the Galerkin approximation with finite elements in weighted norms with uniform and graded meshes. We introduce a discrete decomposition lemma that extends the continuous one and allows us to derive discrete inf-sup conditions in weighted norms. Then, we focus on the convergence of the finite element method. Due to the lack of regularity, the convergence rates are suboptimal for uniform meshes; we show that using graded meshes, optimal rates are recovered. Our theoretical results are supported by several numerical experiments. Finally, we show how our theoretical results apply to certain coupled problems involving fluid flow in porous three-dimensional media with one-dimensional fractures that are found in the analysis of microvascular flows.

Strong and Weak Error Estimates for Elliptic Partial Differential Equations with Random Coefficients

Julia Charrier

SIAM J. Numer. Anal. 50, pp. 216-246 (31 pages)

Online Publication Date: February 09, 2012

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We consider the problem of numerically approximating the solution of an elliptic partial differential equation with random coefficients and homogeneous Dirichlet boundary conditions. We focus on the case of a lognormal coefficient and deal with the lack of uniform coercivity and uniform boundedness with respect to the randomness. This model is frequently used in hydrogeology. We approximate this coefficient by a finite dimensional noise using a truncated Karhunen–Loève expansion. We give estimates of the corresponding error on the solution, both a strong error estimate and a weak error estimate, that is, an estimate of the error commited on the law of the solution. We obtain a weak rate of convergence which is twice the strong one. In addition, we give a complete error estimate for the stochastic collocation method in this case, where neither coercivity nor boundedness is stochastically uniform. To conclude, we apply these results of strong and weak convergence to two classical cases of covariance kernel choices, the case of an exponential covariance kernel on a box and the case of an analytic covariance kernel, yielding explicit weak and strong convergence rates.

On Dimension-independent Rates of Convergence for Function Approximation with Gaussian Kernels

Gregory E. Fasshauer, Fred J. Hickernell, and Henryk Woźniakowski

SIAM J. Numer. Anal. 50, pp. 247-271 (25 pages) | Cited 1 time

Online Publication Date: February 09, 2012

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This article studies the problem of approximating functions belonging to a Hilbert space $\mathcal{H}_d$ with an isotropic or anisotropic translation invariant (or stationary) reproducing kernel with special attention given to the Gaussian kernel $K_d(\boldsymbol{x},\boldsymbol{t}) = \exp\big(-\sum_{\ell=1}^d\gamma_\ell^2(x_\ell-t_\ell)^2\big)$ for all $\boldsymbol{x},\boldsymbol{t}\in\mathbb{R}^d.$ The isotropic (or radial) case corresponds to using the same shape parameters for all coordinates, i.e., $\gamma_\ell=\gamma>0$ for all $\ell$, whereas the anisotropic case corresponds to varying $\gamma_\ell$. The approximation error of the optimal approximation algorithm, called a meshfree or kriging method, is known to decay faster than any polynomial in $n^{-1}$, for fixed $d$, where $n$ is the number of data points. We are especially interested in moderate to large $d$, which in particular arise in the construction of surrogates for computer experiments. This article presents dimension-independent error bounds, i.e., the error is bounded by $Cn^{-p}$, where $C$ and $p$ are independent of both $d$ and $n$. This is equivalent to strong polynomial tractability. The pertinent error criterion is the worst case of such an algorithm over the unit ball in $\mathcal{H}_d$, with the error for a single function given by the ${\mathcal L}_2$ norm whose weight is also a Gaussian which is used to “localize” $\mathbb{R}^d$. We consider two classes of algorithms: (i) using data generated by finitely many arbitrary linear functionals, and (ii) using only finitely many function values. Provided that arbitrary linear functional data is available, we show $p=1/2$ is possible for any translation invariant positive definite kernel. We also consider the sequence of shape parameters $\gamma_d$ decaying to zero like $d^{-\omega}$ as $d$ tends to $\infty$. Note that for large $\omega$ this means that the function to be approximated is “essentially low-dimensional.” Then the largest $p$ is roughly $\max(1/2,\omega)$. If only function values are available, dimension-independent convergence rates are somewhat worse. If the goal is to make the error smaller than $Cn^{-p}$ times the initial $(n=0)$ error, then the corresponding dimension-independent exponent $p$ is roughly $\omega$. In particular, for the isotropic case, when $\omega=0$, the error does not even decay polynomially with $n^{-1}$. In summary, excellent dimension-independent error decay rates are possible only when the sequence of shape parameters decays rapidly.

Function Value Recovery and Its Application in Eigenvalue Problems

Ahmed Naga and Zhimin Zhang

SIAM J. Numer. Anal. 50, pp. 272-286 (15 pages)

Online Publication Date: February 09, 2012

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Function value recovery techniques for linear finite elements are discussed. Using the recovered function and its gradient, we are able to enhance the eigenvalue approximation and increase its convergence rate to $h^{2\alpha}$, where $\alpha > 1$ is the superconvergence rate of the recovered gradient. This is true in both symmetric and nonsymmetric eigenvalue problems.

Finite Sections of Random Jacobi Operators

Marko Lindner and Steffen Roch

SIAM J. Numer. Anal. 50, pp. 287-306 (20 pages)

Online Publication Date: February 14, 2012

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This article is about a problem in the numerical analysis of random operators. We study a version of the finite section method for the approximate solution of equations $Ax=b$ in infinitely many variables, where $A$ is a random Jacobi (i.e., tridiagonal) operator. In other words, we approximately solve infinite second order difference equations with stochastic coefficients by reducing the infinite volume case to the (large) finite volume case via a particular truncation technique. For most of the paper we consider non-self-adjoint operators $A$, but we also comment on the self-adjoint case when simplifications occur.

On Rapid Computation of Expansions in Ultraspherical Polynomials

María José Cantero and Arieh Iserles

SIAM J. Numer. Anal. 50, pp. 307-327 (21 pages)

Online Publication Date: February 14, 2012

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We present an ${\cal O}(N\log_2N)$ algorithm for the computation of the first $N$ coefficients in the expansion of an analytic function in ultraspherical polynomials. We first represent expansion coefficients as an infinite linear combination of derivatives and then as an integral transform with a hypergeometric kernel along the boundary of a Bernstein ellipse. Following a transformation of the kernel, we approximate the coefficients to arbitrary accuracy using the discrete Fourier transform.

Multigrid Preconditioning of Linear Systems for Interior Point Methods Applied to a Class of Box-constrained Optimal Control Problems

Andrei Drăgănescu and Cosmin Petra

SIAM J. Numer. Anal. 50, pp. 328-353 (26 pages)

Online Publication Date: February 28, 2012

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In this article we construct and analyze multigrid preconditioners for discretizations of operators of the form ${\mathcal D}_{\lambda}+{\mathcal K}^*{\mathcal K}$, where $D_{\lambda}$ is the multiplication with a relatively smooth function $\lambda>0$ and ${\mathcal K}$ is a compact linear operator. These systems arise when applying interior point methods to the minimization problem $\min_{u} \frac{1}{2}(|\!|{\mathcal K} u-f|\!|^2 +\beta|\!|u|\!|^2)$ with box-constraints $\underline{u}\leqslant u\leqslant\overline{u}$ on the controls. The presented preconditioning technique is closely related to the one developed by Drăgănescu and Dupont [Math. Comp., 77 (2008), pp. 2001–2038] for the associated unconstrained problem and is intended for large-scale problems. As in that work, the quality of the resulting preconditioners is shown to increase as $h\downarrow 0$, but it decreases as the smoothness of $\lambda$ declines. We test this algorithm on a Tikhonov-regularized backward parabolic equation with box-constraints on the control and on a standard elliptic-constrained optimization problem. In both cases it is shown that the number of linear iterations per optimization step, as well as the total number of finest-scale matrix-vector multiplications, is decreasing with increasing resolution, thus showing the method to be potentially very efficient for truly large-scale problems.

Computing Isolated Singular Solutions of Polynomial Systems: Case of Breadth One

Nan Li and Lihong Zhi

SIAM J. Numer. Anal. 50, pp. 354-372 (19 pages)

Online Publication Date: February 28, 2012

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We present a symbolic-numeric method to refine an approximate isolated singular solution $\hat{\mathbf{x}}=(\hat{x}_{1}, \ldots, \hat{x}_{n})$ of a polynomial system $F=\{f_1, \ldots, f_n\}$, when the Jacobian matrix of $F$ evaluated at $\hat{\mathbf{x}}$ has corank one approximately. Our new approach is based on the regularized Newton iteration and the computation of differential conditions satisfied at the approximate singular solution. The size of matrices involved in our algorithm is bounded by $n \times n$. The algorithm converges quadratically if $\hat{\mathbf{x}}$ is close to the isolated exact singular solution.
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